Henon Maps: Open Problems — Question 14

Definitions and mathematical statement

Write $0=(0,0)\in{\mathbb{C}}^2$ for the origin. The supplied record refers to a map $H_2$ but does not define it. Consequently, the following formalizes the small-cycle property conditionally on specifying the intended map $H_2:{\mathbb{C}}^2\to{\mathbb{C}}^2$.

Using the convention that a cycle is a finite periodic orbit, a cycle of $H_2$ is a set \[ \mathcal O(z)=\{z,H_2(z),\ldots,H_2^{n-1}(z)\}, \] where $n\geq 1$ is an integer and $H_2^n(z)=z$. Here $H_2^0$ is the identity and $H_2^j$ denotes the $j$-fold iterate. Say that $H_2$ has small cycles if every neighborhood $U$ of $0$ in the usual topology of ${\mathbb{C}}^2$ contains an entire cycle disjoint from $\{0\}$. Equivalently, \[ \forall U\subseteq{\mathbb{C}}^2\text{ a neighborhood of }0,\quad \exists n\in{\mathbb{Z}}_{\geq 1}\;\exists z\in{\mathbb{C}}^2:\quad H_2^n(z)=z \quad\text{and}\quad \forall j\in\{0,\ldots,n-1\},\ H_2^j(z)\in U\setminus\{0\}. \] The question is whether the intended $H_2$ has this property. Its identity and any associated hypotheses must be recovered from the source before this becomes a fully specified mathematical problem. The periodic-orbit convention above allows fixed points other than the origin; the supplied excerpt does not specify a different convention for cycles.

Short English statement

For the intended map $H_2$, does every neighborhood of the origin in ${\mathbb{C}}^2$ contain a whole periodic orbit that avoids the origin? The supplied excerpt omits the definition of $H_2$, so the question remains conditional on that missing definition.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $801062$, exact source problem code catalogue entry, titled "Henon Maps: Open Problems --- Question 14." The record attributes the problem to Berger and Bedford, with proposed year 2023. Its status is supplied as open; this label is not an independently verified mathematical or literature assessment.
• Berger and Bedford, Henon Maps: A List of Open Problems (2023), Question 14, https://arxiv.org/abs/2312.03907. This reference and attribution are supplied by the registry background. The recorded access date is 2026-10-16 and the extraction method is arxiv_latex; neither access nor extraction has been independently verified here.